Relativistic Stars in dRGT Massive Gravity (dRGT MG)

Nagoya University @ Japan

Masashi YamazakiIn collaborate with

T. Katsuragawa, S.Nojiri, S. D. Odintsov

Reference:

T. Katsuragawa, S.Nojiri, S. D. Odintsov, MY,PRD 93 124013 (2016)

2/47Contents

n  Point of view for relativistic starsin dRGT massive gravity

n  The Vainshtein mechanism and k-mouflage

n  The dRGT massive gravity and its decoupling limit

n  UV behaviors in dRGT massive gravity

n  Modified TOV eqs. (including hydrostatic eq.)in the theory

n  Future works and summery

3/47A Brief Concept for Massive Gravity

A theory for massive spin-2 particle

Naively, we expect …

The Compton length

GR like behaviors (Effectively massless)

MG behaviors ↓

・Weaken gravitational force ・Graviton condensation

etc.… Radial

coordinate

V ⇠ 1

r

V ⇠ 1

re�mr

4/47Astrophysical PhenomenaGravity theory in astrophysics

(inside the Compton length )General Relativity

[I. Debono, G. F. Smoot (2016)]

⇡

5/47The Neutron Starʼs Inner Structures

[F. Weber, Prog. Part. Nucl. Phys. 54 (2005) 193]

6/47Neutron Starsʼ Maximum Mass

[Demorest et al., Nature 467 (2010) 1081]

A observation result by a binary pulsar

7/47Neutron Stars in F(R) Gravity

0

0.5

1

1.5

2

2.5

3

11 12 13 14 15 16 17

M/M

R (km)

SLy

GR =-0.001 =-0.002 =-0.005 =-0.008

[S. Capozziello et al., (2016)]

Nucleons model

⦿ The maximum mass is drastically changed.

S =

Zd4x

p� det(g)R1+✏

⦿

Mass［

×M⦿ ］

Radius［km］

8/47The Purpose of This Research

This research

In massive gravity, neutron stars’ maximum mass should be the same with GR

for restoring GR behaviors.

Previous researches

By using modified gravity (f(R) gravity), heavier neutron stars are made

for passing observational constraints.

9/47Contents

n  Point of view for relativistic starsin dRGT massive gravity

n  The Vainshtein mechanism and k-mouflage

n  The dRGT massive gravity and its decoupling limit

n  UV behaviors in dRGT massive gravity

n  Modified TOV eqs. (including hydrostatic eq.)in the theory

n  Future work and summery

10/47The degrees of freedom (DOF)

Up to 6 modes [Chamberlin et. al. (2012)]

Gravitational Wave (GW) polarizations in MG

The mode is a ghost DOF. ↓

In ghost-free theory,it’s absent.

The vector modesdon’t couple with matters

The extra DOF makes a fifth force

↓ It should be hidden in short distances.

11/47The Screening Mechanisms

Making the scalar DOF (the breathing mode) not effective in short distance.

There are several ideas to achieve it.

In short distance, the environment ...

n  makes the graviton heavier.

n  suppress the graviton-matter coupling.

n  changes the scalar fieldʼs effective metric drastically.

The massive gravity’s case

12/47The Vainshtein Mechanism

Gravity

PressureThe tensor modes as in GR

The scalar mode (The fifth force)

Non-linear derivative couplings become relevant in short-range. The effects “screen” the scalar DOF.

13/47

The lowest energy that a new interaction with additional DOF in MG

Decoupling Limit and Scalar-Tensor Theories

The Planck Energy

Energy

Energies that new interactions emerge

[The decoupling limit (DL)] Focusing the lowest interaction only. The theory is a scalar-tensor theory.

14/47k-mouflage (EB, Deffayet, Zior ʻ09)

Kinetic screening in scalar-tensor theories

Equation of Motion (EOM) in abstract @2(h+ �) =

T

M2P

, @2h+m2 �KNL

��| {z }⌘E�

= 0

S

k-mouflage

= M

2

P

Zd4x

p�g

⇥R+ �R+m

2

KNL(�, @�, @2

�, . . . )⇤+ Sm[g]

⇠ M

2

P

Z(h@2

h+ �@

2

h+m

2

KKL +O�h

3

�) + hT

Brans-Dicke term

Non-linear derivative couplings

@2�+m2E� =T

M2P

Kinetic Camouflage

15/47The Effect of Non-linear Kinetic Terms

• 

• 

@2�+m2E� =T

M2P

, @2(h+ �) =T

M2P

m2E� ⌧ @2� ⇠ T

M2P

) @2h ⇠ 0

m2E� � @2� ⇠ 0 ) @2h ⇠ T

M2P

GR Restoring

The scalar DOF couples with matters strongly

Non-linear derivative couplings dominate ↓

Non-linear derivative couplings screen the scalar DOF

16/47Contents

n  Point of view for relativistic starsin dRGT massive gravity

n  The Vainshtein mechanism and k-mouflage

n  The dRGT massive gravity and its decoupling limit

n  UV behaviors in dRGT massive gravity

n  Modified TOV eqs. (including hydrostatic eq.)in the theory

n  Future work and summery

17/47The Massless Spin-2 Field Theory in Flat Sp.

n  It is written by a symmetric Lorentz tensor field.

n  Ghosts by higher derivative(the Ostrogradsky ghosts) are absent.

n  In Minkowski spacetime

The linearized Einstein-Hilbert action

L = �M2P

4hµ⌫ E↵�

µ⌫ h↵�

E↵�µ⌫ h↵� = �1

2

⇣⇤hµ⌫ � 2@(µ@↵h

↵⌫) + @µ@⌫h� ⌘µ⌫

�⇤h� @↵@�h

↵��⌘

(The normalization is chosen as usual.)

18/47Nonlinear Completion for The Kinetic Term

L = �M2P

4hµ⌫ E↵�

µ⌫ h↵�

L =M2

P

2

p�gR[g]

The nonlinear completion that the massless spin-2 field couples matters is the Einstein-Hilbert action.

n  Self-interaction consistency POV[Deser 1970]

n  The diffeomorphismʼsnonlinear completion POV[Wald 1986]

gµ⌫ = ⌘µ⌫ + hµ⌫

19/47Nonlinear Completion for Mass Terms

Lmass = �1

8m2M2

P

�h2µ⌫ � h2

�

Nonlinear completion by is ... (non-derivative way)

gµ⌫

gµ⌫gµ⌫ = D, det(g) ?

The trivial quantity or the cosmological constant term

Another rank-2 symmetric tensor is needed!

From the condition that ghosts by higher derivatives are absent,

20/47The Theory Space [Arkani-Hamed et al., 2003]

GC → General Coordinate Invariance Symmetry

x

µ

XA

X

A(xµ)

(M4, gµ⌫)

GC GC’

(M04, fAB)

These are called as the “link fields” or the Stuckelberg fields. These are needed for the pullback.

fµ⌫(x) = @µXA(x)@⌫X

B(x)fAB(X(x))

21/47Non-Linear Fierz-Pauli Terms (NLFP)

There are some candidates for NLFP.

Sint = �1

8m2M2

P

Zd4x

p�fHµ⌫H�⌧ (f

µ�f⌫⌧ � fµ⌫f�⌧ )

Sint = �1

8m2M2

P

Zd4x

p�gHµ⌫H�⌧ (g

µ�g⌫⌧ � gµ⌫g�⌧ )

Hµ⌫ = gµ⌫ � fµ⌫ These are just inverse matrixes of these metric.

Taking decoupling limit

The Gallileon Theories

[Boulware D G and Deser S, 1972]

[Arkani-Hamed et al., 2003]

22/47The Goldstone Expansion

x

µ

XA

X

A(xµ)

(M4, gµ⌫) (M04, fAB)

X

A(x) = X

A0 (x) + ⇡

A(x)

X

A0 (x) ⌘ �

Aµ x

µ⇡

A(x) = �

Aµ (A

µ(x) + ⌘

µ⌫@⌫�)

The scalar-vector decompositionThe identity map

Hµ⌫ =gµ⌫ � fµ⌫

=hµ⌫ � (@µA⌫ + @⌫Aµ)� 2@µ@⌫�� @µA�@⌫A�

� @µ@��@⌫@��� (@⌫A

�@µ@��+ @µA�@⌫@��)

23/47Interaction Scales in The Spin-2 TheoryThe canonical normalizations for fields

hµ⌫ = MPhµ⌫ , Aµ = MPmAµ, � = MPm2�

In the limit of MP ! 1, m ! 0 (m ⌧ MP)

The strongest interaction or

The lowest energy scale interaction

The scalar’s cubic self-interaction

Lint �m2M2P

@µ@⌫m2

�

MP

! @µ@�m2

�

MP

@⌫@�

m2

�

MP

!

⇠ 1

(m4MP)1/5⇥5

⇣@2�

⌘3

This should be constant for DL to remain this coupling.

It leads the Ostrogradsky ghost. (especially BD ghost)

⌘ ⇤5

24/47The dRGT Massive Gravity

eliminates the Ostrogradsky ghost that emerge in nonlinear.

Although, it remains not ghost-like scalar DOF (the fifth force).

S = M

2P

Zd4x

p�g

R�m

2k=4X

k=0

�kek

⇣pg

�1f

⌘!

ek(X) =1

k!XI1

[I1 · · ·XIk

Ik]

cf. e0(X) = 1, e1(X) = Tr(X), e2(X) =1

2

⇣Tr(X)2 � Tr

�X2

�⌘, · · ·

[C. de Rham, G. Gabadadze, and A. J. Tolley (2010)]

25/47The Condition for Flat Sol. In Vacuum

gµ⌫ = fµ⌫ = ⌘µ⌫ )p

g�1f = 1

) Iµ⌫⇣p

g�1f⌘= (�0 + 3�1 + 3�2 + �3)⌘µ⌫ .

Gµ⌫ = Tµ⌫ = 0 ) �0 + 3�1 + 3�2 + �3 = 0.

Gµ⌫ +m20Iµ⌫(

pg�1f,�n) = 2Tµ⌫

There is a relationship between free parameters.

26/47The Strong Coupling Scale in dRGT MG

The special choices of interaction terms rise the lowest strong coupling scale.

m2M2P

h

MP

@2

m2

�

MP

!2

=1

(m2MP)1/3⇥3h⇣@2�

⌘2

This is fixed for DL.

The multiplication of the power of

for the above term is also remained in the DL.

@2

m2

�

MP=

1

(m2MP)1/3⇥3@2�

⌘ ⇤3

27/47DL in The dRGT MG

S =

Zd4x

�1

2h

µ⌫E↵�µ⌫ h↵� + h

µ⌫X

(1)µ⌫ +

↵

⇤33

h

µ⌫X

(2)µ⌫ +

�

⇤63

X

(3)µ⌫ + Tµ⌫ h

µ⌫

!

�µ⌫ = @µ@⌫ �,

X(1)µ⌫ =

1

2✏µ

↵⇢�✏⌫�⇢��↵� ,

X(2)µ⌫ = �1

2✏µ

↵⇢�✏⌫��

��↵��⇢�,

X(3)µ⌫ =

1

6✏µ

↵⇢�✏⌫����↵��⇢����.

↵ ⌘ �1

2(�2 + �3),

� ⌘ 1

2�3

If we choosethe model becomes trivial theory in DL. This is called as the minimal model.

↵ = � = 0 ) �2 = �3 = 0 ) �0 + 3�1 = 0

These can be diagonalized.

28/47The minimal modeln  The scalar and tensor interactions in minimal model

and static and spherical symmetric (SSS) configurations are absent not only in DL but also until the Planck energy.[S. Renaux-Petel (2014)]

n  There is possibility that the system does not have the Vainshtein mechanism because of the absence of higher derivative couplings.

n  The minimal model in non SSS configurationscan have the scalar and tensor interactions.[S. Renaux-Petel (2014)]

The Vainshtein mechanism in the minimal modelshould be checked by making solutions

29/47Contents

n  Point of view for relativistic starsin dRGT massive gravity

n  The Vainshtein mechanism and k-mouflage

n  The dRGT massive gravity and its decoupling limit

n  UV behaviors in dRGT massive gravity

n  Modified TOV eqs. (including hydrostatic eq.)in the theory

n  Future work and summery

30/47The quantum (loop) corrections

n  The Gallileon non-renormalization thm.protects the graviton mass at 1-loop order.(Technically natural graviton mass)

n  The destabilization of the potential is suppressedat 1-loop order.

n  Matter loop structures are the same with GRfor 1-loop order.

n  Graviton loops leads new ops., but itʼs suppressed.

The effective field theory (EFT) description in DL can also be accepted in beyond DL.

The MG models in short distances may be influenced by quantum corrections.

[C. de Rham, L. Heisenberga, and R. H. Ribeiroa (2013)]

31/47The Gallileon Non-Renormalization Thm.

L(�) =5X

i=1

ciLi,

L1 = �, L2 = (@�)2, L3 = @2�(@�)2,

L4 = (@�)2⇥(@2�)2 � (@µ@⌫�)

2⇤,

L5 = (@�)2⇥(@2�)3 + 2(@µ@⌫�)

3 � @2�(@µ@⌫�)2⇤

The coefficients for linear combination are protected w.r.t. quantum corrections.

↓ The Gallileon theories can be treated by tree-level only.

The Gallileon Lagrangian

32/47The Logic for technically natural graviton mass

The interactions      in full theoryare suppressed by the Planck mass for DL interactions.

MG’s quantum corrections vanish in DL.

The Gallileon non-renormalization thm.

�m2 . m2

✓m

MPl

◆2

h2(@2⇡)n

33/47Contents

n  Point of view for relativistic starsin dRGT massive gravity

n  The Vainshtein mechanism and k-mouflage

n  The dRGT massive gravity and its decoupling limit

n  UV behaviors in dRGT massive gravity

n  Modified TOV eqs. (including hydrostatic eq.)in the theory

n  Future work and summery

34/47Metric Ansatz

The reference metric is fixed as flat.�

(gµ⌫dxµdx⌫ = �A(�)dt2 +B(�)d�2 +D(�)2d⌦2

fµ⌫dxµdx⌫ = �dt2 + d�2 + �

2d⌦2

D(�)2 ⌘ r2 ) � ⌘ �(r)8<

:A(�) ⌘ e2⌫(r)

B(�)�0(r)2 ⌘ e2�(r) =⇣1� 2GM(r)

r

⌘�1

8<

:gµ⌫dxµdx⌫ = �e

2⌫(r)dt2 +⇣1� 2GM(r)

r

⌘�1dr2 + r

2d⌦2

fµ⌫dxµdx⌫ = �dt2 + (�0(r))2dr2 + �(r)2d⌦2

This function is a gauge function.�

The Static and Spherical Symmetric (SSS) configuration

35/47The Gauge Function

rµGµ⌫ = rµT

⌫µ = 0

The additional terms come from mass and interaction terms

rµIµ⌫ = 0 (m0 6= 0)

Gµ⌫ +m20Iµ⌫(

pg�1f,�n) = 2Tµ⌫

The equations in SSS becomes a constraint equation that determine the gauge function

36/47The EOMAssuming perfect fluid as matter

Tµ⌫ = (�⇢(r), p(r), p(r), p(r)),

rµTµ⌫ = 0 ) ⌫0(r) = � p0(r)

p(r) + ⇢(r).

GM 0 =4⇡G⇢r2 +m2

0

2r2Itt

p0 =��4⇡Gpr3 +GM �m2

0r3Irr

�(p+ ⇢)

r(r � 2GM)

Modified mass conservation

Substitute this to EOM

Modified hydrostatic equation

37/47The Nontrivial Constraint

� p0

p+ ⇢= ⌫0 =

G

r24⇡pr3 +M � m2

0r3Irr

1� 2GM

r2nd order for χ

4th order for χ�

Substitute

rµIµr

=�0

r2e���⌫

�(�0 + 3(�1 + �2))�

�2� 2e� + �⌫0e⌫

�

� �1

✓2

r+ ⌫0

◆e�� � 2

r

�r2e�+⌫

�2�2

⇥r(1� e�) + �(1� e� + r⌫0)e⌫

⇤

=0

These are set to 0 in the minimal model

Especially, this determines the mass parameter algebraically in the minimal model.

38/47The Equations and DOF in General Case

(M,�; ⇢, p)

p = p(⇢(r))

EOS

(M,�; p)

(⌫,M,�; ⇢, p)

2 independent eqs. → Determine M and p

1 nontrivial eq. → Determine

(Gµ⌫ +m2

0Iµ⌫ = 2Tµ⌫

rµTµ⌫ = 0 = rµIµ⌫

(Gµ⌫ +m2

0Iµ⌫ = 2Tµ⌫

rµIµ⌫ = 0

(Gµ⌫ +m2

0Iµ⌫ = 2Tµ⌫

rµIµ⌫ = 0

�

39/47The EOM in minimal model

q(r) � p�(r)

p(r) + �(r), m(r) =

1

2r � 1

2r

�1 � 1

2q(r)r

��2

,

8�pq + 8�p� � 16�p��

q+ 16�

p�q�

q2

=q

r2+

2

r3+ 3�2 (rgM�)2 q

� 1

r3

�6qq�r3 � 2q3r3 � 2q��r3 + 4q2r2 � 4q�r2 + 5qr + 2

��1 � 1

2qr

��2

+1

r2

�3q�r2 � 3q2r2 + 3qr

�(q�r + q)

�1 � 1

2qr

��3

+1

r(q��r + 2q�) (1 + qr)

�1 � 1

2qr

��3

+3

2r(q�r + q)

2(1 + qr)

�1 � 1

2qr

��4

The mass parameter m(r) can be eliminated from EOM by the new constraint.

Solving 3rd-order ODE for ρ(r) (EOS is used.)

Dimensionless graviton mass

↵ ⌘ m

M�=

p⇤

M�⇠ 10�99

40/47The Boundary Condition

The GR caseThe minimal modelin dRGT MG (not asymptotic flat)

The same radius

for the same central density

Considering differences for the gravitational effects inside the stars

41/47Numerical Method

Solving from the center to outer

Obtaining the radius (The pressure in

there is equal to 0.)

Is the radius the same with GR?

Improving

NO!

YES!

The center of the star（Initial conditions）(⇢(r = 0) = ⇢cP 0(r = 0) = 0

The solution is consistent with BC.

Solving 3rd-order ODE for ρ(r)

⇢00(r = 0)

42/47Results 1: A Quark Star Case

Mass［

×M⦿ ］

Radius［× GM⦿］

GR

The minimal model of dRGT MG

Decreasing the maximum mass

by using MIT bag model

[PRD 93 (2016) 124013]

43/47Results 2: A Traditional Star Case

by using SLy model

Radius［× GM⦿］

Mass［

×M⦿ ］

GR

The minimal model of dRGT MG

[PRD 93 (2016) 124013]

Decreasing the maximum mass

44/47The Summary of These Results

n  The minimal model in the dRGT MGhas smaller neutron starsʼ maximum massthan these in GR.

n  It is caused by the algebraic equationfor mass parameter m(r) that is proper to minimal model.

n  The minimal model in the dRGT MGdoes not have screening mechanisms in these case.

n  In the point of view of observation,the GR solutions is preferred to the minimal modelʼs solutions.Because the large neutron starʼs mass is observed.

45/47Contents

n  Point of view for relativistic starsin dRGT massive gravity

n  The Vainshtein mechanism and k-mouflage

n  The dRGT massive gravity and its decoupling limit

n  UV behaviors in dRGT massive gravity

n  Modified TOV eqs. (including hydrostatic eq.)in the theory

n  Future work and summery

46/47Future Directions

n  Considering models around the minimal model

n How is the parameter choices crucial?

n  Adding perturbations for the breaking SSS configuration

n  Is the Vainshtein mechanism restored by this?

n How strong are perturbations needed?

n It might be uncommon strength in the solar-system

n  (Considering non-minimal models)

47/47Summeryn  Relativistic starsʼ maximum mass is good indicator

for verifying the Vainshtein mechanism.n  The Vainshtein mechanism is caused by

non-linear kinetic terms. (cf. k-mouflage)n  The decoupling limit of dRGT MG has

the Gallileon-type interactions (non-linear kinetic terms).n  The minimal model is the special model

that has not the non-linear derivative couplingbelow the Planck scale.

n  The dRGT MG isnʼt influenced by 1-loop quantum corrections.It can be treated as classical theory effectively.

n  Modified TOV eqs. contains the new constrainsfor fixing the gauge function.

n  The minimal modelʼs maximum mass is smallar thanGRʼs maximum mass.